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In this paper we reveal the existence of a large family of new, nontrivial and Lipschitz traveling waves for the 2D Euler equation at an arbitrarily small distance from the Poiseuille flow in $H^s$, with $s<3/2$, at the level of the vorticity., Comment: 74 pages

We give a simpler proof of Vishik's nonuniqueness Theorem for the forced 2D Euler equation in the vorticity class $L^1\cap L^p$ with $2

In this paper we consider the incompressible 2D Euler equation in an annular domain with non-penetration boundary condition. In this setting, we prove the existence of a family of non-trivially smooth time-periodic solutions at an arbitrarily small distance from the stationary Taylor-Couette flow in $H^s$, with $s<3/2$, at the vorticity level., Comment: 76 pages

One way to understand the deformation theory of a tensor category $M$ is through its Davydov-Yetter cohomology $H_{DY}^{\ast}(M)$ which in degree 3 and 4 is known to control respectively first order deformations of the associativity coherence of $M$ and their obstructions. \\ In this work we take the task of developing an analogous theory for the deformation theory of tensor triangulated categories with a focus on derived categories coming from algebraic geometry. We introduce the concept of perfect pseudo dg-tensor structure $\Gamma$ on an appropriate dg-category $\mathscr{T}$ as a truncated dg-lift of a tensor triangulated category structure on $H^{0}(\mathscr{T})$ and we define a double complex $DY^{\ast,\ast}(\Gamma)$ and we see that the 4th cohomology group $HDY^{4}(\Gamma)$ of the total complex of $DY^{\ast,\ast}(\Gamma)$ contains information about infinitesimal first order deformations of the tensor structure., Comment: Article draws extensively from author's PhD thesis, found in HAL tel-04018741

We investigate maximal potential energy dissipation as a selection criterion for subsolutions (coarse grained solutions) in the setting of the unstable Muskat problem. We show that both, imposing this criterion on the level of convex integration subsolutions, and the strategy of Otto based on a relaxation via minimizing movements, lead to the same nonlocal conservation law. Our main result shows that this equation admits an entropy solution for unstable initial data with an analytic interface., Comment: 63 pages; updated to accepted version

Let $X$ be a smooth projective variety over $\mathbb{C}$ with big (anti-)canonical bundle. It is known that in this situation the Balmer spectrum of the tensor triangulated category of perfect complexes $Perf(X)$ of $X$ equipped with the derived tensor product $\otimes_{X}^{\mathbb{L}}$ recovers the space $X$. In this work we study the possible tensor triangulated category structures one can put on $Perf(X)$. As an application we prove a monoidal version of the well-known Bondal-Orlov reconstruction theorem.

In this paper we reveal the existence of a large family of new, nontrivial and smooth traveling waves for the 2D Euler equation at an arbitrarily small distance from the Couette flow in $H^s$, with $s<3/2$, at the level of the vorticity. The speed of these waves is of order 1 with respect to this distance. This result strongly contrasts with the setting of very high regularity in Gevrey spaces (see arXiv:1306.5028), where the problem exhibits an inviscid damping mechanism that leads to relaxation of perturbations back to nearby shear flows. It also complements the fact that there not exist nontrivial traveling waves in the $H^{\frac{3}{2}+}$ neighborhoods of Couette flow (see arXiv:1004.5149)., Comment: 60 pages

We consider smooth solutions of the Burgers-Hilbert equation that are a small perturbation $\delta$ from a global periodic traveling wave with small amplitude $\epsilon$. We use a modified energy method to prove the existence time of smooth solutions on a time scale of $\frac{1}{\epsilon\delta}$ with $0<\delta\ll\epsilon\ll1$ and on a time scale of $\frac{\epsilon}{\delta^2}$ with $0<\delta\ll\epsilon^2\ll1$. Moreover, we show that the traveling wave exists for an amplitude $\epsilon$ in the range $(0,\epsilon^*)$ with $\epsilon^*\sim 0.23$ and fails to exist for $\epsilon>\frac{2}{e}$., Comment: Some typos and minor errors fixed

We construct mixing solutions to the incompressible porous media equation starting from Muskat type data in the partially unstable regime. In particular, we consider bubble and turned type interfaces with Sobolev regularity. As a by-product, we prove the continuation of the evolution of IPM after the Rayleigh-Taylor and smoothness breakdown exhibited in [18,17]. At each time slice the space is split into three evolving domains: two non-mixing zones and a mixing zone which is localized in a neighborhood of the unstable region. In this way, we show the compatibility between the classical Muskat problem and the convex integration method., Comment: 40 pages, 3 figures

We consider classical solutions of the inviscid Surface Quasi-geostrophic equation that are a small perturbation $\epsilon$ from a radial stationary solution $\theta=|x|$. We use a modified energy method to prove the existence time of classical solutions from $\frac{1}{\epsilon}$ to a time scale of $\frac{1}{\epsilon^4}$. Moreover, by perturbing in a suitable direction we construct global smooth solutions, via bifurcation, that rotate uniformly in time and space., Comment: 25 pages

We obtain new semiclassical estimates for pseudodifferential operators with low regular symbols. Such symbols appear naturally in a Cauchy Problem related to recent weak solutions to the unstable Muskat problem constructed via convex integration in [CCF16]. In particular, our new estimates reveal the tight relation between the speed of opening of the mixing zone and the regularity of the interphase., Comment: 24 pages

We study the formation of singularities for the Euler-Alignment system with influence function $\psi=\frac{k_\alpha}{|x|^\alpha}$ in 1D. As in [20] the problem is reduced to the analysis of a nonlocal 1D equation. We show the existence of singularities in finite time for any $\alpha$ in the range $0<\alpha<2$ in both the real line and the periodic case., Comment: 26 pages

We prove the existence of infinitely many mixing solutions for the Muskat problem in the fully unstable regime displaying a linearly degraded macroscopic behaviour inside the mixing zone. In fact, we estimate the volume proportion of each fluid in every rectangle of the mixing zone. The proof is a refined version of the convex integration scheme presented in [DS10, Sze12] applied to the subsolution in [CCF16]. More generally, we obtain a quantitative h-principle for a class of evolution equations which shows that, in terms of weak*-continuous quantities, a generic solution in a suitable metric space essentially behaves like the subsolution. This applies of course to linear quantities, and in the case of IPM to the power balance $\mathbf{P}$ (14) which is quadratic. As further applications of such quantitative h-principle we discuss the case of vortex sheet for the incompressible Euler equations., Comment: 28 pages

We consider the 2D Boussinesq equations with a velocity damping term in a strip $\mathbb{T}\times[-1,1]$, with impermeable walls. In this physical scenario, where the \textit{Boussinesq approximation} is accurate when density/temperature variations are small, our main result is the asymptotic stability for a specific type of perturbations of a stratified solution. To prove this result, we use a suitably weighted energy space combined with linear decay, Duhamel's formula and "bootstrap" arguments., Comment: 40 pages. Revised argument in section 6.2, results unchanged. arXiv admin note: text overlap with arXiv:1804.08490; text overlap with arXiv:1410.1415 by other authors

In this paper, we consider a confined physical scenario to prove global existence of smooth solutions with bounded density and finite energy for the inviscid incompressible porous media (IPM) equation. The result is proved using the stability of stratified solutions, combined with an additional structure of our initial perturbation, which allows us to get rid of the boundary terms in the energy estimates., Comment: 24 pages. Updated funding

The variability of young stellar objects is mostly driven by star-disk interactions. In long-term photometric monitoring of the accreting T Tauri star GI Tau, we detect extinction events with typical depths of $\Delta V \sim 2.5$ mag that last for days-to-months and often appear to occur stochastically. In 2014 - 2015, extinctions that repeated with a quasi-period of 21 days over several months is the first empirical evidence of slow warps predicted from MHD simulations to form at a few stellar radii away from the central star. The reddening is consistent with $R_V=3.85\pm0.5$ and, along with an absence of diffuse interstellar bands, indicates that some dust processing has occurred in the disk. The 2015 -- 2016 multi-band lightcurve includes variations in spot coverage, extinction, and accretion, each of which results in different traces in color-magnitude diagrams. This lightcurve is initially dominated by a month-long extinction event and return to the unocculted brightness. The subsequent light-curve then features spot modulation with a 7.03 day period, punctuated by brief, randomly-spaced extinction events. The accretion rate measured from $U$-band photometry ranges from $1.3\times10^{-8}$ to $1.1\times10^{-10}$ M$_\odot$ yr$^{-1}$ (excluding the highest and lowest 5% of high- and low- accretion rate outliers), with an average of $4.7 \times 10^{-9}$ M$_\odot$ yr$^{-1}$. A total of 50% of the mass is accreted during bursts of $>12.8\times10^{-9}$ M$_\odot$ yr${^{-1}}$, which indicates limitations on analyses of disk evolution using single-epoch accretion rates., Comment: 15 pages, 17 figures

In this paper, we show the existence of a family of compactly supported smooth vorticities, which are solutions of the 2D incompressible Euler equation and rotate uniformly in time and space., Comment: 58 pages

We prove the existence of mixing solutions of the incompressible porous media equation for all Muskat type $H^5$ initial data in the fully unstable regime. The proof combines convex integration, contour dynamics and a basic calculus for non smooth semiclassical type pseudodifferential operators which is developed., Comment: 73 pages, 2 figures. This version includes the case of variable opening of the mixing zone and emphasizes the semiclassical analysis viewpoint

In this paper, we show the existence of the first non trivial family of classical global solutions of the inviscid surface quasi-geostrophic equation., Comment: 85 pages, 2 figures, approx. 5.000 lines of code

We show that there exists a family of analytic convex global rotating solutions for the vortex patch equations, bifurcating from ellipses. As a byproduct, the analyticity proof can also be adapted to the rotating patch solutions bifurcating from disks (also known as V-states) for both the Euler and the generalized surface quasi-geostrophic equation., Comment: 31 pages

In this paper, we prove the existence of smooth initial data for the 2D free boundary incompressible Navier-Stokes equations, for which the smoothness of the interface breaks down in finite time into a splash singularity., Comment: 82 pages, 9 figures